Category Archives: Games

Recall the game “matching pennies“. Player I has to chose between ‘0’ or ‘1’, player II has to chose between ‘0’ and ‘1’.No player knows what is the choice of the other player before making his choice. Player II pays to player I one dollar if the two number match and gets one dollar from player I if they do not match.

The optimal way to play the game for each of the player is via a mixed strategy. Player I has to choose ‘0’ with probability 1/2 and ‘1’ with probability 1/2. And so is player II.

Now, suppose that the game is the same except that if the outcomes match and both player chose ‘1’ player II pays 5 dollars to player I and not one dollar. (All other payoffs remain at one dollar.)

Test your intuition: What is the probability player I should play ‘1’. (More than 1/2? less than 1/2? more than 1/3? less than 1/3? more than 2/3? less than 2/3?)

The issue of “what is gambling” is very intereting. In an earlier post entitled “Chess can be a game of luck” the interesting question regarding “games of luck” and “games of skill” was discussed. It was argued that for a betting game, if, over time, for all players (or even for most players), the expected gains are negative, then we can regard the game as primarily a game of luck. (The claim is that this is a reasonable interpretation of the current law, even if not the only possible interpretation.)

A) We have a chess tournament where each of forty chess players pay 50 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.

B) We have a chess tournament where each of forty chess players pay 20,000 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.

Before dealing with these two rather realistic scenarios let us consider the following more hypothetical situations.

C) Suppose that chess players have a quality measure that allows us to determine the probability that any one player will beat the other. Two players play and bet. The strong player bets 10 dollars and the waek player bets according to the probability he will win. (So the expected gain of both player is zero.)

D) Suppose again that chess players have a quality measure that allows us to determine the probability that any one players will beat the other. Two players play and bet. The strong player bets 100,000 dollars and the weak player bets according to the probability he will wins. (Again, the expected gain of both players is zero.)

When we analyze scenarios C and D the first question to ask is “What is the game?” In my opinion we need to consider the entire setting, so the “game” consists of both the chess itself and the betting around it. In cases C and D the betting aspects of the game are completely separated from the chess itself. We can suppose that the higher the stakes are, the higher the ingredient of luck of the combined game. It is reasonable to assume that version C) is mainly a game of skill and version D) is mainly a game of luck.

Now what about the following scenarios:

E) Two players play chess and bet 5 dollars.

Here the main ingredient is skill; the bet only adds a little spice to the game.

F) Two players play chess and bet 100,000 dollars.

Well, to the extent that such a game takes place at all, I would expect that the luck factor will be dominant. (Note that scenario F is not equivalent to the scenario where two players play, the winner gets 300,000 dollars and the loser gets 100,000 dollars.)

Let us go back to the original scenarios A) and B). Here too, I would consider the ingredients of luck and skill to be strongly dependant on the stakes. The setting of scenario A) can be quite compatible with a game of skill where the prizes give some extra incentives to participants (and rewards for the organizers), while in scenario B) it stands to reason that the luck/gambling factor will be dominant.

One critique against my opinion is: What about tennis tournaments where professional tennis players are playing on large amounts of prize money? Are professional tennis tournaments games of luck? There is one major difference between this example and examples A and B above. In tennis tournaments there are very large prizes but the expected gain for a player is positive, all (or at least most)players can make a living by participating. This changes entirely the incentives. This is also the case for various high level professional chess tournaments.

For mathematicians there are a few things that sound strange in this analysis. The luck ingredient is not invariant under multiplying the stakes by a constant, and it is not invariant under giving (or taking) a fixed sum of money to the participants before the game starts. However, these aspects are crucial when we try to analyze the incentives and motives of players and, in my opinion, it is a mistake to ignore them.

The main part of the lecture was about the four old theorems in the table above and about what should replace the two question marks. The left side of the table deals with properties of the majority voting rule for binary preferences. The right side of the table is about general voting rules. On the top tight is the famous Arrow Impossibility Theorem. The table is filled by two theorems I proved in 2002 (in this paper) and it now looks like this:Continue reading →

Well, I wrote an article (in Hebrew) about it in the Newspaper Haaretz. An English translation appeared in the English edition. Here is an appetizer:

During World War II, many fighter planes returned from bombing missions in Japan full of bullet holes. The decision was made to reinforce the planes, and their natural tendency was to bolster the hardest-hit sections in the body of the plane. However, the mathematician George Dantzig suggested that it was precisely the parts that were hit less that needed to be armored. Was he right?

Months after all the commentators described Hillary Clinton’s chances as so slim she was bound to lose her campaign for the Democratic presidential nomination, she continued to fight for her candidacy, saying she believed she would win and keeping up her attack on her rival. Did she act rationally? And did Benjamin Netanyahu and Tzipi Livni act rationally when each declared victory on election night? Did Meretz supporters who voted for Kadima act rationally? Is there an election method in which it would be rational for all voters to vote in accordance with their genuine preferences?

Aumann and Myerson proposed that if political and ideological matters are put aside, the party forming the coalition would (or should) prefer to form the coalition in which its own power (according to the Shapley-Shubik power index) is maximal. They expected that this idea would have some predictive value — even in reality, where political and ideological considerations are of importance. A few days ago Yair Tauman, another well-known Israeli game theorist, mentioned on TV this recipe as a normative game-theoretic recommendation in the context of the recent Israeli elections. (For Yair’s analysis see also this article. (I even sent a critical comment.))

Over the years, Aumann was quite fond of this suggestion and often claimed that in Israeli elections it gives good predictions in some (but not all) cases. The original paper mentions the Israeli 1977 elections and how delighted one of the authors was that four months after the elections a major “centrist” party joined the coalition, leading to a much better Shapley value for the party forming the coalition.

I was quite skeptical about the claim that the maximum-power-to-the-winning-party rule has any predictive value and in 1999 with the help of Sergiu Hart I decided to test this claim. I asked Aumann which Israeli coalition he regards as fitting his prediction the best. His answer was the 1988 election where Shamir’s party, the Likud, had a very large Shapley value in the coalition it formed. We checked how high the Shapley value was compared to a random coalition that the winning party could have formed. Continue reading →

The problem.

OK, we had an election and have a new parliament with 120 members. The president has asked the leader of one party to form a coalition. (This has not happened yet in the Israeli election but it will happen soon.) Such a coalition should include parties that together have more than 60 seats in the parliament.

Can game theory make some prediction as to which coalition will be formed or give some normative suggestions on which coalition to form?

Robert Aumann and Roger Myerson made (in 1977) the following concrete suggestion. (Update: A link to the full 1988 paper is now in place.) The party forming the coalition would (or should) prefer to form the coalition in which its own power (according to the Shapley-Shubik power index) is maximal. Of course Continue reading →